Finite Strains of a Granular Material

  • Oxana Sadovskaya
  • Vladimir Sadovskii
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 21)


A mathematical model of developed flow of a granular material is considered. On the phenomenological level, elastic properties characteristic for a compacted material and viscous properties appearing in loosening are taken into account. Exact solutions of problems on rotational and plane-parallel motion of a material with stagnant zones are constructed. Using them, influence of viscosity on a flow pattern is analyzed.


Variational Inequality Granular Material Constitutive Relationship Shear Angle Stagnant Zone 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Anand, L.: On H. Hencky’s approximate strain-energy function for moderate deformations. Transactions of ASME. J. Appl. Mech. 46(1), 78–82 (1979)CrossRefGoogle Scholar
  2. 2.
    Bagnold, R.A.: Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear. Proc. R. Soc. Ser. A 225(1160), 49–63 (1954)CrossRefGoogle Scholar
  3. 3.
    Bakhvalov, N.S., Zhidkov, N.P., Kobelkov, G.M.: Chislennye Metody (Numerical Methods). BINOM, Moscow (2003)Google Scholar
  4. 4.
    Batchelor, G.K.: Introduction to Fluid Dynamics. Cambridge University Press, Cambridge, UK (1967)Google Scholar
  5. 5.
    Berdichevsky, V.L.: Variational Principles of Continuum Mechanics, vol. 1: Fundamentals. Springer, Berlin (2009)Google Scholar
  6. 6.
    Berdichevsky, V.L.: Variational Principles of Continuum Mechanics, vol. 2: Applications. Springer, Berlin (2009)Google Scholar
  7. 7.
    Dolgunin, V.N., Borschev, V.Y.: Bystrye Gravitaczionnye Techeniya Zernistykh Materialov: Tekhnika Izmereniya, Zakonomernosti, Tekhnologicheskoe Primenenie (Fast Gravity Flows of Granular Materials: Technique of Measurement, Regularities, Technological Application). Mashinostroenie-1, Moscow (2005)Google Scholar
  8. 8.
    Gantmacher, F.R.: The Theory of Matrices, vol. 1–2. AMS Chelsea Publishing, Providence, Rhode Island, USA (1990)Google Scholar
  9. 9.
    Godunov, S.K.: Elementy Mekhaniki Sploshnoi Sredy (Elements of Continuum Mechanics). Nauka, Moscow (1978)Google Scholar
  10. 10.
    Goldshtik, M.A.: Proczessy Perenosa v Zernistom Sloe (Transfer Processes in Granular Layer). Inst. Teplofiziki SO RAN, Novosibirsk (1984)Google Scholar
  11. 11.
    Golovanov, Y.V., Shirko, I.V.: Review of current state of the mechanics of fast motions of granular materials. In: Shirko, I.V. (ed.) Mechanics of Granular Media: Theory of Fast Motions, Ser. New in Foreign Science, vol. 36, pp. 271–279. Mir, Moscow (1985)Google Scholar
  12. 12.
    Goodman, M.A., Cowin, S.C.: Two problems in the gravity flow of granular materials. J. Fluid Mech. 45(2), 321–339 (1971)CrossRefGoogle Scholar
  13. 13.
    Grigorian, S.S.: On basic concepts in soil dynamics. J. Appl. Math. Mech. 24(6), 1604–1627 (1960)CrossRefGoogle Scholar
  14. 14.
    Joseph, D.: Stability of Fluid Motions, Springer Tracts in Natural Philosophy, vol. 27–28. Springer, New York (1976)Google Scholar
  15. 15.
    Kondaurov, V.I., Nikitin, L.V.: Teoreticheskie Osnovy Reologii Geomaterialov (Theoretical Foundations of Rheology of Geomaterials). Nauka, Moscow (1990)Google Scholar
  16. 16.
    Korobeinikov, S.N.: Nelineinoe Deformirovanie Tverdykh Tel (Nonlinear Deformation of Solids). Izd. SO RAN, Novosibirsk (2000)Google Scholar
  17. 17.
    Lehmann, T., Guo, Z., Liang, H.: The conjugacy between Cauchy stress and logarithm of the left stretch tensor. Eur. J. Mech. A/Solids 10(4), 395–404 (1991)Google Scholar
  18. 18.
    Marchuk, G.I.: Methods of Numerical Mathematics. Springer, Berlin (1975)Google Scholar
  19. 19.
    Maslennikova, N.N., Sadovskii, V.M.: Modeling of the dilatancy under finite strains of granular material. Vestnik Krasnoyarsk. Univ.: Fiz.-Mat. Nauki 4, 215–219 (2005)Google Scholar
  20. 20.
    Meyers, A., Schieße, P., Bruhns, O.T.: Some comments on objective rates of symmetric Eulerian tensors with application to Eulerian strain rates. Acta Mechanica 139(1–4), 91–103 (2000)CrossRefGoogle Scholar
  21. 21.
    Nikolaevskii, V.N.: Governing equations of plastic deformation of a granular medium. J. Appl. Math. Mech. 35(6), 1017–1029 (1971)CrossRefGoogle Scholar
  22. 22.
    Revuzhenko, A.F.: Mekhanika Uprugoplasticheskikh Sred i Nestandartnyi Analiz (Mechanics of Elastic-Plastic Media and Nonstandard Analysis). Izd. Novosib. Univ., Novosibirsk (2000)Google Scholar
  23. 23.
    Revuzhenko, A.F.: Mechanics of Granular Media. Springer, Berlin (2006)Google Scholar
  24. 24.
    Revuzhenko, A.F., Stazhevskii, S.B., Shemyakin, E.I.: On mechanism of deformation of granular material under large shears. Fiz.-Tekhn. Probl. Razrab. Pol. Iskop. 3, 130–133 (1974)Google Scholar
  25. 25.
    Sadovskaya O.V., Sadovskii, V.M.: Rheological models of granular medium under small strains. In: Proceedings of the International Conference on Fundamental and Applied Problems of Mechanics, Izd. Khabarovsk. Gos. Tekhn. Univ., Khabarovsk, vol. 1, pp. 95–108 (2003)Google Scholar
  26. 26.
    Sadovskaya, O.V., Sadovskii, V.M.: The theory of finite strains of a granular material. J. Appl. Math. Mech. 71(1):93–110 (2007)Google Scholar
  27. 27.
    Sadovskii, V.M.: Radial expansion of a granular medium in spherical and cylindrical layers. J. Appl. Mech. Tech. Phys. 50(3), 519–524 (2009)CrossRefGoogle Scholar
  28. 28.
    Savage, S.B.: Gravity flow of cohesionless granular materials in chutes and channels. J. Fluid Mech. 92(1), 53–96 (1979)CrossRefGoogle Scholar
  29. 29.
    Shkadov, V.Y., Zapryanov, Z.D.: Techeniya Vyazkoi Zhidkosti (Flows of Viscous Fluid). Izd. Mosc. Univ., Moscow (1984)Google Scholar
  30. 30.
    Sokolovskii, V.V.: Statics of Granular Media. Pergamon Press, Oxford (1965)Google Scholar
  31. 31.
    Tritenko, A.N.: Mekhanika Sypuchei Sredy (Mechanics of Granular Medium). Izd. Vologod. Univ., Vologda (2005)Google Scholar
  32. 32.
    Xiao, H., Bruhns, O.T., Meyers, A.: Logarithmic strain, logarithmic spin and logarithmic rate. Acta Mechanica 124(1–4), 89–105 (1997)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.ICM SB RASKrasnoyarskRussia

Personalised recommendations